If the line \( \displaystyle\frac{x-1}{l}=\frac{y-2}{m}=\frac{z+1}{n} \) passes through \( (-1,0,1) \), then its direction cosines \( l,m,n \) are:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{\sqrt{3}},\ \frac{1}{\sqrt{3}},\ -\frac{1}{\sqrt{3}} \)
The line passes through (1, 2, −1), read from its equation, and through the given point (−1, 0, 1).
Its direction ratios are the differences of the coordinates:
\( \displaystyle (-1-1,\ 0-2,\ 1-(-1))=(-2,-2,2) \)
\( \displaystyle (-2,-2,2)\propto(1,1,-1) \)
Dividing by \( \sqrt{1^{2}+1^{2}+(-1)^{2}}=\sqrt{3} \) gives the direction cosines:
\( \displaystyle l,m,n=\frac{1}{\sqrt{3}},\ \frac{1}{\sqrt{3}},\ -\frac{1}{\sqrt{3}} \)